I'm huge into metal music and have some metal-related tattoos on the torso. They are unlikely to offend anyone, but they are 'death' themed and therefore might not agree so well with Taiwanese officials, given as I would likely be teaching mostly young kids. I'm a pretty good teacher with experience and am planning to travel to Taipei in early August to interview with a variety of schools.

Of course there's no reason a school would need to know about this, as the artwork is not visible under a shirt. But I wonder, would the medical check involve inspection of the torso (abdomen, back, chest)?

A chest X-ray for TB requires that nothing more than a plain cloth cover the upper body, but wearing a simple undershirt should eliminate the need to disrobe.

Would any other procedure during the exam require removing one's shirt?

The check up varies by time and place, although you shouldn't have to remove your shirt and if you did it shouldn't be a problem. The health check is largely an inconvenience more than something useful for anyone. If you have measle vaccination certificates, take them with you or pay extra an waste more time.

The only time you'd have to disrobe (taking off your shirt) is for the chest xray. And even that, if you do your health check in a city hospital, will be covered by an unflattering, long green smock that will extend to your knees.

Now, if you're going to stay for your APRC they will do a skin test and that's all clothes off - head to toe - checking for sores.

Anyway loads of Taiwanese guys have tattoos with all kinds of themes. The school I worked for just asked us that if we did have tattoos to try to keep them covered, but they weren't too bothered about it. It certainly isn't going to keep you from passing your medical check. They didn't bat an eye at my tongue piercing so I'm sure you'll be fine.